WS9 - 2 cos x(b Write all solutions to the diﬀerential...

Math 152 Workshop 9 Spring 2011 1. A small object of unknown temperature was placed in a large room that had the ﬁxed temperature 30 ◦ C. After 10 minutes, the object’s temperature is-10 ◦ C, and after an additional 10 minutes, the object’s temperature was-5 ◦ C. What was the initial temperature of the object? (Assume that the temperature obeys Newton’s law of cooling: the rate of change of the temperature of the object is proportional to the diﬀerence in temperature between the object and the constant room temperature.) 2. Find a solution of y0 = y 1-x 2 which passes through (0 , 1). Write the solution explicitly as y = f ( x ). Graph the solution curve. What is the domain of the function describing the solution curve? 3. If a function f is continuous on the interval [ a, b ) and if lim x → b-f ( x ) = + ∞ or-∞ then we say “ f explodes at b .” (a) Consider the following functions on an interval [0 , b ) with b > 0. For each function, ﬁnd b so that the function explodes at b . Use your calculator to show graphically what occurs: x 2 +1 x-1 ; cos x x-2 ; x-

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Unformatted text preview: 2 cos x . (b) Write all solutions to the diﬀerential equation y = y 2 subject to the initial condition y (0) = y . Can you ﬁnd a solution that explodes at 10? And another one that explodes at 5? Can you ﬁnd a solution that explodes at x when x > 0? How does the initial condition at x = 0 connect with a speciﬁed explosion at x ? Graph one exploding solution. 4. A tissue culture grows until it has an area of 9 cm 2 . Let A ( t ) be the area of the tissue at time t . One model for the growth rate is A ( t ) = k q A ( t ) ± 9-A ( t ) ² for some constant k . This is reasonable because the number of cells on the edge is proportional to q A ( t ) and most of the growth occurs on the edge. (a) Without solving the equation, show that the maximum rate of growth occurs at any time when A ( t ) = 3 cm 2 . (b) Assume that k = 6. Find the solution corresponding to A (0) = 1 and sketch its graph. (c) Do the same for A (0) = 4....
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